Lecture 9

Motivation

• Many real-world data sets vary in both space and time.

  • Examples: rainfall, temperature, air pollution, crop yield, disease spread.

• We often want to model how similarity (correlation) changes across both dimensions.

Complexity of spatio-temporal models

• additional dependencies: in space and time

• spatial and temporal behaviour can be independent on each other, or dependent: i.e. properties of the spatial structure vary through time (or vice versa)

Spatio temporal models for areal data

\(\frac{\text{Number of cases } Y_{st}}{\text{Expected numer of cases }E_{st}}\) in Ohio from 1968 to 1988

Space

Spatio temporal models for areal data

\(\frac{\text{Number of cases } Y_{st}}{\text{Expected numer of cases }E_{st}}\) in Ohio from 1968 to 1988

Time

Spatio temporal models for areal data

\(\frac{\text{Number of cases } Y_{st}}{\text{Expected numer of cases }E_{st}}\) in Ohio from 1968 to 1988

Time

Spatio temporal models for areal data

The observation model \[ Y_{st}|\lambda_{st}\sim\text{Poisson}(E_{st}\lambda_{st}) \]

We are going to see different models for the linear predictor \(\eta_{st} = \log \lambda_{st}\)